code: def kernel_pdf_lognorm_(x, sample, bw):
      """Log-normal kernel for density, pdf, estimation, explicit formula.
  
      Jin, Kawczak 2003
      """
      term = 8 * np.log(1 + bw)  # this is 2 * variance in normal pdf
      pdf = (
          1
          / np.sqrt(term * np.pi)
          / sample
          * np.exp(-((np.log(x) - np.log(sample)) ** 2) / term)
      )
      return pdf.mean(-1)
  
imports: import numpy
  from scipy import special, stats
label_desc: Use image node_id 25 for calling a global function with example usage: kernel_pdf_lognorm_(x, sample, bw) and returns: pdf